Notes on Liouville Theory at c<=1

TL;DR

This paper explores Liouville conformal field theory at central charge c<=1, examining its mathematical structure, potential physical interpretations involving timelike bosons, and connections to minimal models and string amplitudes.

Contribution

It provides new insights into the analytic structure of Liouville theory at c<=1, including the three-point function and the spectrum restrictions, extending previous work on c=1 and minimal models.

Findings

The three-point function has a unique analytic factor matching minimal models.

A non-analytic factor yields a well-defined metric under certain conditions.

The c=1 case aligns with a non-rational CFT limit of minimal models.

Abstract

The continuation of the Liouville conformal field theory to c<=1 is considered. The viability of an interpretation involving a timelike boson which is the conformal factor for two-dimensional asymptotically de Sitter geometries is examined. The conformal bootstrap leads to a three-point function with a unique analytic factor which is the same as that which appears along with the fusion coefficients in the minimal models. A corresponding non-analytic factor produces a well-defined metric on fields only when the central charge is restricted to those of the topological minimal models, and when the conformal dimensions satisfy h>(c-1)/24. However, the theories considered here have a continuous spectrum which excludes the degenerate representations appearing in the minimal models. The c=1 theory has been investigated previously using similar techniques, and is identical to a non-rational CFT which arises as a limit of unitary minimal models. When coupled to unitary matter fields, the non-unitary theories with c<=-2 produce string amplitudes which are similar to those of the minimal string.